Optimizing Elliptic Curve Scalar Multiplication for Small Scalars

Pascal Giorgi 1, * Laurent Imbert 2, 1 Thomas Izard 1
* Corresponding author
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : On an elliptic curve, the multiplication of a point P by a scalar k is defined by a series of operations over the field of definition of the curve E, usually a finite field Fq. The computational cost of [k]P = P + P + * * * + P (k times) is therefore expressed as the number of field operations (additions, multiplications, inversions). Scalar multiplication is usually computed using variants of the binary algorithm (double-and-add, NAF, wNAF, etc). If s is a small integer, optimized formula for [s]P can be used within a s-ary algorithm or with double-base methods with bases 2 and s. Optimized formulas exists for very small scalars (s 5). However, the exponential growth of the number of field operations makes it a very difficult task when s > 5. We present a generic method to automate transformations of formulas for elliptic curves over prime fields in various systems of coordinates. Our method uses a directed acyclic graph structure to find possible common subexpressions appearing in the formula and several arithmetic transformations. It produces efficient formulas to compute [s]P for a large set of small scalars s. In particular, we present a faster formula for [5]P in Jacobian coordinates. Moreover, our program can produce code for various mathematical software (Magma) and libraries (PACE).
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Pascal Giorgi, Laurent Imbert, Thomas Izard. Optimizing Elliptic Curve Scalar Multiplication for Small Scalars. Mathematics for Signal and Information Processing, 2009, San Diego, CA, United States. pp.74440N, ⟨10.1117/12.827689⟩. ⟨lirmm-00424282⟩

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